It has been shown that the electrical field gradients created in OET can be numerically modeled by using the Finite Element Method (FEM) [2
]. Here the electric fields within the OET chamber are modeled and the resulting gradient of the square of the electric field is used to find the DEP force using the well-know equation [6
is the permittivity of the medium, Re[k
)] is the Clausius-Mossotti factor and E2
is the gradient of the electrical field squared [6
]. The Clausius-Mossotti factor is given by;
are the complex permittivities of the particle and medium respectively and ω is the angular frequency. The relation between the real permittivity and complex permittivity is given by ε*
) with ε
the permittivity and σ is the conductivity. Whilst this approach gives good agreement between experimentally measured forces and the simulated forces for small homogeneous particles the modeling of cells is more complicated as the structure of a cell is more complicated. A reasonable approximation can be made by modeling the cell as a single shell of low conductivity material surrounding a highly conductive core [15
Dielectrophoretic force can either be positive (pDEP), towards areas of high electric field, or negative (nDEP), away from these areas depending on the sign of the Clausius-Mossotti factor. A cell will experience nDEP at low frequencies and pDEP at high frequencies crossing over at a certain frequency. This crossover frequency is dependent on the physical properties of the suspending medium and the cell. Some of the properties can be deduced by measuring this frequency in liquid of varying conductivity [16
]. Cultured HeLa cells were removed from a substrate with trypsin before being re-suspended in an isotonic sugar solution [1
]. By adding varying amounts of cell culture medium to the solution the conductivity was varied from 3.2 to 2.2×10−2
and the crossover frequency was measured as shown in . The range of conductivities that can be explored is limited by the response of the OET chamber with higher conductivities giving a reduction in the force [14
]. The trap stiffness experiments shown earlier in this paper were performed in 1×10−2
media at 100kHz giving a positive force.
Fig. 4 Simulations of DEP. The experimentally measured crossover frequency is plotted as points along side simulated lines for a conductive core (radius 7.5 μm, relative permittivity 50, and conductivity 0.53Sm−1) surrounded by a single shell (more ...)
These experimental results were compared to a single shell model in MATLAB. The effect of the cell’s insulating shell is accounted for by defining an effective permittivity for the core-shell structure that is given by [15
are the complex permittivities of the core and shell, respectively, and r1
are their radii. This effective permittivity can then be used in place of the particle’s permittivity in Equation 2
], allowing us to calculate the Clausius-Mossotti factor. Using values for the permittivity and conductivity of the cell’s cytoplasm and membrane from the literature (cytoplasm εr
= 50 σ = 0.53 Sm−1
, membrane, εr
= 7 σ = 1 μSm−1
]), the crossover frequency for varying medium conductivities can be modeled. It was found that to fit the simulations with the experimentally measured crossover frequencies a membrane conductivity of 0.87 μSm−1
and thickness of 5nm gave the best agreement. This is consistent with value from the literature where the membrane thickness has been measured as 4.97±0.2nm by scanning electron microscopy [17
]. These values give us a Clausius-Mossotti factor of 0.8 at 1×105
Hz, the frequency the experiments were performed at. To calculate the force it is now just necessary to find E2
. 3D numerical simulations were performed in COMSOL Multiphysics (COMSOL) with the light spot being modeled as a saturated Gaussian that matches the profiles measured from the optical spots from the experiments. These saturated Gaussians are entered into COMSOL as profiles in conductivity of the a-Si varying from 1×10−6
the dark and illuminated conductivities of the a-Si, respectively (see ). These are the conductivities measured for an illumination power of 2.5Wcm−2
, the intensity produced by our data projector when reduced through a 20x objective.
Fig. 5 Measured light patterns and simulated forces. A) The profile of the optical spots are plotted as crosses and compared to saturated Gaussian curves plotted as solid lines for optical spot diameters 73 μm (blue), 49 μm (red), 31 μm (more ...)
The size and shape of the optical profiles were measured by recording video of the trap and analyzing a frame in a graphics package (National Instruments Vision Assistant). As the images of the traps were projected in red, the red pixel value is used as a measure of the projected light intensity. The optical profiles are shown in and it can be seen that the measured optical profiles match well with saturated Gaussian curves. It has been shown that the conductivity of the photoconductor is proportional to the illumination intensity [14
shows the experimental results for the 31 μm diameter trap plotted against the simulated forces at different heights from the a-Si surface. As the cells have a diameter of 15 μm, one might expect the simulated forces at 7.5 μm from the surface to match the experimental data. However it was found that the shapes of the experimental and simulated traps at this height did not match well.
at 7.5 μm above the a-Si surface is effectively assuming that this value does not vary over the volume of the cell or that the higher forces near the OET surface are cancelled by the lower forces further from the a-Si so that the value half way up the cell is a good average. This is not a valid assumption in this case as the cell is large compared to the distance over which the gradients vary and the force profile closer to the surface is different from the profile further from the a-Si so that the two effects do not cancel each other. From it can be seen that the force profile at 7.5 μm from the surface is close to an ideal trap profile where the force is proportional to the displacement, a trap that could be represented with a constant spring constant over the central part of the trap. This means that once the trap is moved a small amount the trapped particle would immediately feel a restoring force pushing it towards the center of the trap. The experimental results however show a trap that is completely flat in the centre and the particle does not experience a force until it is several microns from the centre of the trap. This is similar to the force profiles lower down in the trap, closer to the a-Si. Here E2
is greater so to compare the trap profiles the magnitude of the gradients has to be reduced. It was found that the profile at 3.5 μm above the a-Si matched the measured trap profiles surprisingly well when reduced by a factor of four for all four traps, as shown in . As the cell is not visibly deformed the extra force can’t be due to the cell being pulled closer to the surface although this may have a small effect. This factor of four is simply the amount that the force at 3.5 μm from the a-Si is greater than the average force over the volume of the cell. The simplicity of this modeling method gives us an incredibly useful tool for future experiments allowing us to accurately predict the force on the cell and hence design light patterns that will translate into the patterns of force we desire. One such desirable pattern is a simple ideal trap with a constant spring constant as mentioned earlier. shows that this can be produced by reducing the size of the optical spot to 12 μm. Although reducing the illuminated area to this extent does reduce the maximum force that can be exerted by 17%, the stiffness is the same. In this case the trap stiffness is 3×10−6
and the maximum velocity that can be achieved is 50 μms−1
with the 12 μm trap.
An analytical expression can be fitted to the numerically simulated force profile. It was found that a 3rd order polynomial could be fitted with good agreement. This expression can then be integrated to find the work done by the trap on the particle which can then be used to find the potential energy. shows the potential energy landscape that the particle experiences due to the trap. The depth of the potential well is found to be 1.6×10−16J (4×104 kBT).
Potential energy well. The potential energy well experienced by the trapped HeLa cell is calculated from the force verses position profile for the 12 μm diameter trap. The depth of this potential energy well is 1.6×10−16 J.